Sure, let's analyze the problem step-by-step:
### Step 1: Define the Sample Space
The sample space [tex]\(n(S)\)[/tex] represents the total number of possible outcomes when flipping a coin 12 times. Each flip of the coin has 2 possible outcomes: heads (H) or tails (T).
For 12 flips, the total number of possible outcomes is:
[tex]\[ n(S) = 2^{12} \][/tex]
Given the calculation:
[tex]\[ n(S) = 4096 \][/tex]
### Step 2: Define the Desired Outcome [tex]\(n(A)\)[/tex]
We are interested in finding the probability of getting exactly 6 heads in 12 flips.
The number of ways to choose 6 heads out of 12 flips can be calculated using the combination formula [tex]\( \binom{n}{k} \)[/tex], which represents the number of ways to choose [tex]\(k\)[/tex] successes (heads) out of [tex]\(n\)[/tex] trials (flips):
[tex]\[ \binom{12}{6} = \frac{12!}{6! \cdot (12-6)!} \][/tex]
Given the calculation:
[tex]\[ n(A) = 924 \][/tex]
### Step 3: Calculate the Probability [tex]\(P(A)\)[/tex]
The probability [tex]\(P(A)\)[/tex] of getting exactly 6 heads in 12 flips is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[ P(A) = \frac{n(A)}{n(S)} \][/tex]
Given the values:
[tex]\[ P(A) = \frac{924}{4096} = 0.2255859375 \][/tex]
### Summary
- Total Sample Space [tex]\( n(S) \)[/tex] = 4096
- Number of favorable outcomes [tex]\( n(A) \)[/tex] = 924
- Probability [tex]\( P(A) \)[/tex] = 0.2255859375
So, the probability of getting exactly 6 heads when flipping a coin 12 times is approximately 0.2256 or 22.56%.
